Because \(\sigma _{int}\)  depends only on the band structure, we carried out the first-principles calculations to have a better understanding of the large \(\sigma _{xy}^{int}\)  in LiMn\(_6\) Sn\(_6\) . Figure REF  (a) shows the band structure without SOC in the vicinity of \(E\rm _F\) . Both spin-up and spin-down bands pass through the \(E\rm _F\) , with several crossings close to the \(E\rm _F\) . If the SOC is included into account, the band structure is almost unaltered due to the light elements, only with some small gaps opened in the band crossings. This band structure is consistent with the two types of carriers revealed by the Hall measurements. It is worth pointing out that there is a spin-polarized linear crossing slightly above the \(E\rm _F\)  at the K point, and it is gapped with the consideration of SOC. This is reminiscent of the Chern gapped Dirac fermions found in TbMn\(_6\) Sn\(_6\)  [1]}. Figure REF (c) shows the spin-resolved density of states near the \(E\rm _F\) . The bands near the \(E\rm _F\)  are mainly hybridized by the Mn-\(d\)  and Sn-\(p\)  orbitals. The DOS for both directions have relatively small values at the \(E\rm _F\) , consistent with the several band crossings near the \(E\rm _F\)  revealed by the band structure. The calculated magnetic moment based on the spin-resolved DOS is 2.4 \(\mu _B\) /Mn, same with the value obtained in the magnetization curves. The energy dependence of intrinsic AHC \(\sigma _{xy}^{int}\)  can be calculated from Berry curvature based on the band calculations. As shown in Fig. REF  (b), there is a peak near the \(E\rm _F\)  with the value at \(E\rm _F\)  of 300 \(\Omega ^{-1}\)  cm\(^{-1}\) , which is close to the experimental value. In the time-reversal-symmetry-broken systems, a band crossing near the \(E\rm _F\)  with a SOC induced gap is believed to contribute a large Berry curvature [2]}. Thus, the peak of \(\sigma _{xy}^{int}\)  near \(E\rm _F\)  can be understood as the consequence of the close positions of the band crossings to the \(E\rm _F\) . We can also notice a higher peak of \(\sigma _{xy}^{int}\)  centered at about 0.15 eV, which is followed by a deep valley around 0.2 eV. If we assume the \(R\) Mn\(_6\) Sn\(_6\)  compounds have similar band structure, the \(E\rm _F\)  can be lifted for about 0.23 eV by the 2 extra valence electrons of \(R\) . This results in the smaller \(\sigma _{xy}^{int}\)  in \(R\) Mn\(_6\) Sn\(_6\) , as shown by the upper dashed line in Fig. REF (b). All the calculation results fit well with the experimental observations, suggesting the reliability of our calculations. Based on the calculations, a much larger AHC up to 0.73 \(e^2/h\)  per Mn layer can be expected if the ferromagnetic compounds with \(E\rm _F\)  located at about 0.15 eV can be synthesized, such as (Li, \(R\) )Mn\(_6\) Sn\(_6\)  and (Mg, \(R\) )Mn\(_6\) Sn\(_6\) .
